高级解决方案中概率与数论问题.pdfVIP

1.Davidflipsafaircoinfivetimes.Computetheprobabilitythatthefourthcoinflipisthefirst

coinflipthatlandsheads.

1

Answer:

16

4

11

Solution:Davidmustflipthreetails,thenheads.Thishappenswithprobability=.

216

2.Findthelargestintegerthatdividesp2−1forallprimesp3.

Answer:24

Solution:Forp=5,p2−1=24andhencetheanswermustbeadivisorof24.Since

2

p−1=(p+1)(p−1)andallprimesp3areodd,oneof(p+1)and(p−1)willbedivisibleby

2andtheotherwillbedivisibleby4.Sotheansweriseither8or24.Furthermore,sincep3

cannotbedivisibleby3,either(p+1)or(p−1)mustbedivisibleby3.Hence,theansweris

24.

3.Wesaythatanumberisarithmeticallysequencedifthedigits,inorder,formanarithmetic

sequence.Computethenumberof4-digitpositiveintegerswhicharearithmeticallysequenced.

Answer:30

Solution:Thereare9numberswithanarithmeticsequenceofdifference0(1111through9999).

Thereare6withanarithmeticsequenceofdifference1(1234through6789).Thereare3with

anarithmeticsequenceofdifference2(1357through3579).Thereare7withanarithmetic

sequenceofdifference−1(3210through9876).Thereare4withanarithmeticsequenceof

difference−2(6420through9753),andthereis1withadifferenceof−3(9630).Theansweris

therefore9+6+3+7+4+1=30.

4.Foranypositiveintegerx≥2,definef(x)tobethepr